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   Introduction Introduction
   What's the essence? Essence
   Memorizing Memorizing
   Understanding Understanding
   Summarizing Summarizing
   Examples Examples

Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
General ideas

   Exercises Exercises
   Final thoughts Final thoughts
Page 1 | Page 2 | Proof(s)

Example 6--Page 2

We want 6_6201.gif (2574 bytes)6_6202.gif (1581 bytes). Looking at the right side we want to get 6_6203.gif (1239 bytes) from the left side. We know by the Product to Sum formulas, sine multiplied by cosine will become sine plus sine, and 2pi_o_7.gif (1024 bytes), 3pi_o_7.gif (1014 bytes)r_arrow.gif (902 bytes)5pi_o_7.gif (998 bytes), pi_o_7.gif (938 bytes); 3pi_o_7.gif (1014 bytes), pi_o_7.gif (938 bytes)r_arrow.gif (902 bytes)2pi_o_7.gif (1024 bytes), 4pi_o_7.gif (1016 bytes). So we will get 6_6203.gif (1239 bytes) and 6_6204.gif (1224 bytes), but by using the Derived formula, 6_6205.gif (1293 bytes)6_6206.gif (1570 bytes)6_6203.gif (1239 bytes). We reach our goal.

6_6207.gif (11101 bytes)

Since 6_6208.gif (3009 bytes).

6_6209.gif (4404 bytes)

Now for the second factor, we still didn't reach the goal 6_6210.gif (1167 bytes), and cancel 6_6211.gif (1338 bytes). But we see it is simpler than the last step. Same idea, we want 6_6212.gif (1842 bytes)6_6213.gif (1704 bytes). It looks like the Sum to Product formulas. If we use the Sum to Product formula of sines immediately, we have 6_6212.gif (1842 bytes)6_6214.gif (1825 bytes). The right side of it is not 6_6215.gif (1666 bytes). We need pi_o_7.gif (938 bytes), not 3pi_o_14.gif (1046 bytes)! How can we get pi_o_7.gif (938 bytes)? If we use the Derived formulas we have 6_6212.gif (1842 bytes)6_6216.gif (1832 bytes)6_6217.gif (1781 bytes)6_6218.gif (1710 bytes), which is what we want.

6_6219.gif (9104 bytes)

The identity has been proven.

This problem is a little bit more difficult than the previous ones. You can see that Derived formulas are very useful for actual angle identities.


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Copyright 1998 LWR, ThinkQuest team 17119. All rights reserved.